For the inner product kernel \(k(x, y) = \varrho \left(\frac{x^\top y}{d}\right)\) , it is shown in [1]} that the kernel matrix \(k(\mathcal {X}, \mathcal {X})\)  can be approximated by the linear combination of all-ones matrix \(1 1^\top \) , sample covariance matrix \(\mathcal {X} \mathcal {X}^\top \)  and identity matrix. To obtain non-asymptotic results on the spectra of the kernel matrix \(k(\mathcal {X}, \mathcal {X})\) , we borrow the technique from [2]}, and show the result for subgaussian entries in the next lemma.
